1. Department of Computer and IT Engineering University of Kurdistan
Social Network Analysis
Centrality
By: Dr. Alireza Abdollahpouri

2. Power in social networks
Certain positions within the network give nodes more power
Directly affect/influence others
Control the flow of information
Avoid control of others

3. Centrality in social networks
Centrality encodes the relationship between structure and power in groups
Certain positions within the network give nodes more power or importance
How do we measure importance?
Who can directly affect/influence others?
Highest degree nodes are “in the thick of it”
Who controls information flow?
Nodes that fall on shortest paths between others can disrupt the flow of information between them
Who can quickly inform most others?
Nodes who are close to other nodes can quickly get information to them

4. Measures and Metrics
Knowing the structure of a network, we can calculate various useful quantities or measures that capture particular features of the network topology.
basis of most of such measures are from social network analysis
Degree distribution, Average path length, Density
Centrality
Degree, Eigenvector, Katz, PageRank, Hubs, Closeness, Betweenness,
Several other graph metrics
Clustering coefficient, Assortativity, Modularity, …

5. Characterizing networks: Who is most central?

6. Network centrality Which nodes are most ‘central’?
Definition of ‘central’ varies by context/purpose
Local measure:
degree
Relative to rest of network:
closeness, betweenness, eigenvector (Bonacich power centrality), Katz, PageRank, …
How evenly is centrality distributed among nodes?
Centralization, hubs and authorities, …

7. Centrality: who’s important based on their network position
In each of the following networks, X has higher centrality than Y according to a particular measure
indegree
outdegree
betweenness
closeness

8. He who has many friends is most important.
Degree centrality (undirected)
He who has many friends is most important.
When is the number of connections the best centrality measure?
people who will do favors for you
people you can talk to (influence set, information access, …)
influence of an article in terms of citations (using in-degree)

9. divide by the max. possible, i.e. (N-1)
Degree: normalized degree centrality
divide by the max. possible, i.e. (N-1)

10. Degree centrality The number of others a node is connected to
Node with high degree has high potential communication activity 1 2 3 4 5
node
In-degree
Out-degree
Total degree 1 2 3 5 4 1 2 3 4 5

11. Extensions of undirected degree centrality - prestige
indegree centrality
a paper that is cited by many others has high prestige
a person nominated by many others for a reward has high prestige

12. How much variation is there in the centrality scores among the nodes?
Centralization: how equal are the nodes?
How much variation is there in the centrality scores among the nodes?
Freeman’s general formula for centralization:
(can use other metrics, e.g. gini coefficient or standard deviation)
maximum value in the network

13. Degree Centrality in Social Networks
Freeman: 0.0
Variance: 0.0
Freeman: 1.0
Variance: 3.9
Freeman: .02
Variance: .17
Freeman: .07
Variance: .20

14. Centralization= Σ(C*-Ci) / Max Σ(C*-Ci)
Degree centralization examples
CD = 0.167
CD = 1.0
CD = 0.167
Centralization= Σ(C*-Ci) / Max Σ(C*-Ci)

15. Degree centralization examples
example financial trading networks
high centralization: one node trading with many others
low centralization: trades are more evenly distributed

16. When degree isn’t everything
In what ways does degree fail to capture centrality in the following graphs?
ability to broker between groups
likelihood that information originating anywhere in the network reaches you…

17. Betweenness: another centrality measure
intuition: how many pairs of individuals would have to go through you in order to reach one another in the minimum number of hops?
who has higher betweenness, X or Y? Y X

18. Betweenness centrality
Number of shortest paths (geodesics) connecting all pairs of other nodes that pass through a given node
Node with highest betweenness can potentially control or distort communication 1 2 3 4 5
12
123
124
1245
23
24
245
32
34
35 1 2 3 4 5
452
4523
45
52
523
524

19. Betweenness on toy networks
non-normalized version: A B C D E
A lies between no two other vertices
B lies between A and 3 other vertices: C, D, and E
C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E)
note that there are no alternate paths for these pairs to take, so C gets full credit

20. betweenness centrality: definition
paths between j and k that pass through i
betweenness of vertex i
all paths between j and k
Where gjk = the number of geodesics connecting j-k, and
gjk = the number that actor i is on.
Usually normalized by:
number of pairs of vertices excluding the vertex itself
For directed graph: (N-1)*(N-2)

21. betweenness on toy networks
non-normalized version:

22. betweenness on toy networks
non-normalized version:
broker

23. betweenness on toy networks
non-normalized version:
why do C and D each have betweenness 1?
They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit:
½+½ = 1
Can you figure out why B has betweenness 3.5 while E has betweenness 0.5? C A E B D

24. Betweenness centrality
If you add lines from C to D and from D to H, you remove
the high betweenness centrality of E and F

25. Centrality vs. Centralization
Centrality is a characteristic of an actor’s position in a network
Centralization is a characteristic of a network
Centralization indicates:
- how unequal the distribution of centrality is in a network or
- how much variance there is in the distribution of centrality in a network
Centrality is a micro-level measure
Centralization is a macro-level measure

26. Betweenness Centralization (examples)

27. Comparison Nodes are sized by degree, and colored by betweenness.
What about high degree but relatively low betweenness?
Can you spot nodes with high betweenness but relatively low degree?

28. Extending betweenness centrality to directed networks
We now consider the fraction of all directed paths between any two vertices that pass through a node
paths between j and k that pass through i
betweenness of vertex i
all paths between j and k
Only modification: when normalizing, we have (N-1)*(N-2) instead of (N-1)*(N-2)/2, because we have twice as many ordered pairs as unordered pairs

29. Directed geodesics
A node does not necessarily lie on a geodesic from j to k if it lies on a geodesic from k to j j k

30. closeness: another centrality measure
What if it’s not so important to have many direct friends?
Or be “between” others
But one still wants to be in the “middle” of things,
not too far from the center

31. Closeness centrality: definition
Closeness is based on the length of the average shortest path between a vertex and all vertices in the graph
Closeness Centrality:
depends on inverse distance to other vertices
Normalized Closeness Centrality

32. closeness centrality: toy exampleB C D E

33. closeness centrality: more toy examples

34. Closeness Centrality (examples)
Distance Closeness normalized
Distance Closeness normalized

35. Distance Closeness normalized
Closeness Centrality in Social Networks
Distance Closeness normalized

36. Closeness Centrality in Social Networks
Distance Closeness normalized

37. how closely do degree and betweenness correspond to closeness?
number of connections
denoted by size
closeness
length of shortest path to all others
denoted by color

38. Closeness centrality Values tend to span a rather small dynamic range
typical distance increases logarithmically with network size
In a typical network the closeness centrality C might span a factor of five or less
It is difficult to distinguish between central and less central vertices
a small change in network might considerably affect the centrality order
Alternative computations exist but they have their own problems

39. Influence range
The influence range of i is the set of vertices who are reachable from the node i

40. Extensions of undirected closeness centrality
closeness centrality usually implies
all paths should lead to you
paths should lead from you to everywhere else
usually consider only vertices from which the node i in question can be reached

41. Eigenvector Centrality
Idea: A central actor is connected to other central actors
A natural extension of the degree centrality
For a given graph G:=(V,E) with |V| number of vertices let A be the adjacency matrix. The centrality score of vertex v can be defined as:

42. Eigenvector Centrality
 Google's PageRank is a variant of the eigenvector centrality measure
Node B is more popular in the network if we only extend our vision out to a distance of 1 from each node. But A is connected to nodes that are connected to many other nodes, while B is connected to less-popular nodes. A has a higher eigenvector centrality.

43. Degree vs. Eigenvector Centrality

44. Comparing centralization of different networks
Empirical study:
Comparing centralization of different networks
Comparison of centralization metrics across three networks:
• butland ppi: binding interactions among 716 yeast proteins
• addhealth9: friendships among 136 boys
• tribes: positive relations among 12 NZ tribes

45. Empirical study:
Comparing centralization of different networks
The protein network looks visually centralized, but
• most centralization is local;
• globally, somewhat decentralized.
The friendship network has small degree centrality (why?).
The tribes network has one particularly central node.

46. Centrality in Social Networks
There are other options, usually based on generalizing some aspect of those above:
Random Walk Betweenness (Mark Newman). Looks at the number of times you would expect node I to be on the path between k and j if information traveled a ‘random walk’ through the network.
Peer Influence based measures (Friedkin and others). Based on the assumed network autocorrelation model of peer influence. In practice it’s a variant of the eigenvector centrality measures.
Subgraph centrality. Counts the number of cliques of size 2, 3, 4, … n-1 that each node belongs to. Reduces to (another) function of the eigenvalues. Similar to influence & information centrality, but does distinguish some unique positions.
Fragmentation centrality – Part of Borgatti’s Key Player idea, where nodes are central if they can easily break up a network.
Moody & White’s Embeddedness measure is technically a group-level index, but captures the extent to which a given set of nodes are nested inside a network
Removal Centrality – effect on the rest of the (graph for any given statistic) with the removal of a given node. Really gets at the system-contribution of a particular actor.

47. Questions